We study the monotonicity of the ratios of two Abelian integrals closed integral(gamma i(h)) ydx \ closed integral(gamma i)(h) xydx over three period annuli {gamma(i)(h)}, for i = 1, 2, 3, defined by a seventh-degree hyperelliptic Hamiltonian H(x, y) = y(2) + Psi(x) with a parameter. The parameter makes the problem more challenging to analyze. To overcome the difficulty, we apply some criterion with the help of transformations, tools in computer algebra such as boundary polynomial theory to determine the monotonicity of the ratios. Our results establish the existence and uniqueness of limit cycle bifurcated from each period annulus.

In this paper, we study the monotonicity of the ratio of the Abelian integrals, ∮γi(h)ydx and ∮γi(h)xydx, in an interval, where i=1,2, and γi(h) is a compact component of some hyperelliptic curves with genus 2 as h∈γi(h). We give positive answers to the two conjectures proposed by Wang et al. (2014) [18].

In this paper, we study the monotonicity of the ratio of two Abelian integralsI0(h)=∮γ(h)ydx,andI1(h)=∮γ(h)xydx in an interval Σ, where γ(h) is a compact component of hyperelliptic curves with genus 2 as h∈Σ. We first give the topological classification of these hyperelliptic curves. Then we show what kinds of compact components in the classification ensure that the ratio of the two Abelian integrals is monotone.

In this paper we study three classes of complete hyperelliptic integrals of the first kind, which are some degenerate subfamilies of a family considered in the work of Gavrilov and Iliev. It is shown that the three classes of complete hyperelliptic integrals are Chebyshev, and the exact bounds on the number of zeros of these Abelian integrals are one. This result reveals that there exist degenerate subfamilies of ovals of the hyperelliptic Hamiltonian which are not exceptional families proposed by Gavrilov and Iliev, but the corresponding complete hyperelliptic integrals of the first kind still satisfy the Chebyshev property.